1. No category

EXAMPLES SHEET, FINITE DIMENSIONAL NORMED SPACES
TOM SANDERS
Exercises with daggers (:) are harder, which is not to say that the others are not.
Answers and comments on some of the questions appear at the end.
1. Show that if X is a Banach space and X ˚ is separable and its elements separate points
(i.e. if x ‰ y P X then there is some φ P X ˚ such that φpxq ‰ φpyq) then X is separable.
2. Show that cc is dense in `p for 1 ď p ă 8.
3.
Write Ban1 for the category of Banach spaces in which the morphisms are short
maps. As per Example 2.2 of the notes, write
ιqÑp : `q Ñ `p ; x ÞÑ x,
and show that the direct limit of the system x`q , ιqÑp y over the directed set r1, 8q under
the usual order is (isometrically isomorphic to) c0 .
4. Show that `21 is isometrically isomorphic to `28 over R. (Here `np is the vector space
Rn endowed with the norm
˜
¸1{p
n
ÿ
}x}`np “
|xi |p
i“1
with the usual convention for p “ 8.)
5. Show that if φ : X Ñ Y is a surjective linear isometry from X to Y then it is an
isometric isomorphism.
6. Show that `˚1 is isometrically isomorphic to `8 .
7. : Show that for 1 ď p ă q ă 8 the spaces `p and `q are not isometrically isomorphic.
8.
Show that c and c0 are not isometrically isomorphic.
9. Prove the sequential Banach-Alaoglu theorem (Theorem 2.9) directly by the method
of Proposition 2.8.
10.
Prove that a linear operator T : Y ˚ Ñ X ˚ is weak-* to weak-* continuous if and
only if there is some S P LpX, Y q such that T “ S ˚ .
Last updated : 2nd June, 2015.
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TOM SANDERS
11. Show that LpX, Y ˚ q is metrisable in the topology of pointwise convergence if and
only if X and Y are finite dimensional.
12. Give an example of functions pfn qn and f in `p (1 ă p ă 8) such that φpfn q Ñ φpf q
for all φ P `˚p (we say that fn Ñ f in the weak topology) but fn Û f in `p -norm.
13.
Prove Schur’s property that if φpfn q Ñ φpf q for every φ P `˚1 , then fn Ñ f in
`1 -norm.
14. Show that ∆ and D28 from §3.3 of the notes are homeomorphic.
15. Prove that if T is a compact metrisable space and t P T then there are bounded
continuous functions pfn,t qn such that fn,t Ñ 1ttu pointwise i.e. fn,t Ñ 1ttu in the bounded
pointwise topology.
16. Prove the closed graph theorem for topological spaces: suppose X is a topological
space, Y is a compact Hausdorff space, and f : X Ñ Y . Then f is continuous if and only
if the graph of f , that is the set tpx, f pxqq : x P Xu is closed as a subset of X ˆ Y .
17. Give an example of continuous linear maps Ψ : Cpr0, 1sq Ñ Cpr0, 1sq and Φ :
Cpr0, 1sq Ñ Cpr0, 1sq satisfying the hypotheses of the Robust Banach-Stone theorem (Theorem 3.10)) that are not multiples of isometries i.e. where }Ψ}}Φ} ‰ 1.
18. Give an example of an isometric isomorphism from c0 into Cr0, 1s.
19. : Suppose that Ψ : X Ñ Y is a surjective isometry with Ψp0X q “ 0Y . Show that Ψ
is linear.
20. Prove Lemma 4.5.
21. Prove Lemma 4.6.
22. Show that X ‘ Y with the maps px, yq ÞÑ x and px, yq ÞÑ y is a product in TopVect;
and with the maps x ÞÑ px, 0Y q and y ÞÑ p0X , yq is a coproduct in TopVect.
23. Suppose that X and Y are Banach spaces over a field F. Explain why, despite the
apparently greater number of morphisms in TopVect, their product in Ban1 is a product
in TopVect, and similarly for their coproduct. Hence conclude that this product and
coproduct are continuously isomorphic. (This provides another proof of the last part of
Lemma 4.5, at least up to the specific values of the norms.)
24. Show that `p pXq is a Banach space.
25. Prove Lemma 4.9.
26. Show that the map defined in Example 4.10 really is an isometric isomorphism.
EXAMPLES SHEET, FINITE DIMENSIONAL NORMED SPACES
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27. Check that you believe Lemma 4.12.
28. Show that the map in Exercise 4.14 really is a short projection.
29. Show that if X is a Banach space and φ : X Ñ X has }φ ´ I} ă 1 in the operator
norm then φ is invertible. Hence, or otherwise, show that the set of invertible maps is open
in LpX, Xq endowed with the topology induced by the operator norm.
30. Verify the triangle inequality for the Banach-Mazur distance i.e. show that dBM pX, Zq ď
dBM pX, Y qdBM pY, Zq. Verify that it is also symmetric.
31. Suppose that φ : `1 Ñ c0 is a continuous linear map. Show that ker φ is not
complemented in `1 .
?
32. : Show that if dpX,
`n1 q ď K then there is a subspace Y ď X with dim Y “ k :“ t nu
?
such that dpY, `k1 q ď K.
33. Show that if X is an n-dimensional Banach space and x P X then there is some φ P X ˚
such that φpxq “ }φ}}x}. Hence show that ΦX is an isometric isomorphism between X
and X ˚ ˚.
34. Show that if X is an n-dimensional Banach space then X ˚ is an n-dimensional Banach
space.
35. For i P t0, 1u defined two norms on c0 as follows
¸1{2
˜
8
ÿ
|2´n xn`i |2
.
}x}i :“ sup |xn | `
n
n“1
Show that pc0 , } ¨ }i q is a Banach space. By considering the maps
Tn : pc0 , } ¨ }0 q Ñ pc0 , } ¨ }1 q; px1 , . . . q ÞÑ pxn , x1 , . . . , xn´1 , xn`1 , . . . q
or otherwise show that dBM ppc0 , } ¨ }0 q, pc0 , } ¨ }1 qq “ 1, but that these spaces are not
isometrically isomorphic.
36. Show that if dBM pX ˚ , Y ˚ q “ dBM pX, Y q if X and Y are finite dimensional.
37.
Show that `2 is isometrically isomorphic to L2 pr0, 1sq as claimed in Example 6.1.
More generally use the Gramm-Schmidt process to show that any infinite dimensional
separable Hilbert space is isometrically isomorphic to `2 .
38. Show that On acts transitively on S n´1 .
39. Show that the norm on X P SubpΩq
}X}SG1 :“ suptp´1{2 }X}Lp pPq : p ě 1u,
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TOM SANDERS
is, indeed, a norm and that it is equivalent to the norm } ¨ }SG defined in lectures.
40. Show that if X is a random variable such that Pp|X| ą tq ď expp´ct2 q whenever
t ą C then X is pOC,c p1q, c{2q-sub-exponential.
41. Show that X is sub-exponential if and only if E|X p | “ Oppq for all p ě 1.
Acknowledgements
Thanks to Sean Eberhard and Przemyslaw Mazur for comments and corrections.
Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, United Kingdom
E-mail address: [email protected]